Final answer:
- The projectile leaves the barrel of the cannon with a speed of approximately 0.91 m/s.
- At the point where the friction force is equal to 0.391 N, the ball has maximum speed.
Step-by-step explanation:
To find the speed at which the projectile leaves the barrel of the cannon, we can use the principles of physics.
1. --Calculate the work done by the spring:
The work done by the spring is equal to the potential energy stored in the compressed spring.
The formula for potential energy is:
Potential energy (U) = (1/2) * force constant (k) * displacement²
Given:
- Force constant of the spring (k) = 7.95 N/m
- Displacement of the spring (x) = 4.92 cm = 0.0492 m
Potential energy (U) = (1/2) * 7.95 N/m * (0.0492 m)² = 0.0094 J
--Calculate the work done against friction:
The work done against friction can be calculated using the formula:
Work done against friction (W-friction) = friction force * displacement
Given:
- Friction force (f) = 0.0322 N
- Displacement of the ball in the barrel (d) = 15.4 cm = 0.154 m
Work done against friction (W-friction) = 0.0322 N * 0.154 m = 0.00496 J
-- Calculate the initial kinetic energy of the ball:
The initial kinetic energy of the ball can be found by subtracting the potential energy and the work done against friction from the total energy.
Total energy = Potential energy - Work done against friction
Total energy = 0.0094 J - 0.00496 J = 0.00444 J
-- Calculate the speed of the projectile:
The initial kinetic energy of the ball is equal to the final kinetic energy, given by:
Initial kinetic energy (K-initial) = (1/2) * mass * velocity²
Given:
- Mass of the ball (m) = 5.35 g = 0.00535 kg
- (1/2) * 0.00535 kg * velocity² = 0.00444 J
Simplifying the equation, we find
Velocity² = (0.00444 J) / (0.00535 kg) = 0.829 m²/s²
Taking the square root of both sides, we find:
Velocity = √(0.829 m²/s²) ≈ 0.91 m/s
Therefore, the projectile leaves the barrel of the cannon with a speed of approximately 0.91 m/s.
2)-- Calculate the point of maximum speed:
The point of maximum speed occurs when the acceleration of the ball is zero. In this case, the acceleration becomes zero when the friction force is equal to the force provided by the spring.
Friction force (f) = Force provided by the spring (F-spring)
Given:
- Force constant of the spring (k) = 7.95 N/m
- Force provided by the spring (F-spring) = k * displacement
F-spring = 7.95 N/m * 0.0492 m = 0.391 N
Therefore, at the point where the friction force is equal to 0.391 N, the ball has maximum speed.
Your question is incomplete, but most probably the full question was:
A toy cannon uses a spring to project a 5.35-g soft rubber ball. the spring is originally compressed by 4.92 cm and has a force constant of 7.95 n/m. when the cannon is fired, the ball moves 15.4 cm through the horizontal barrel of the cannon, and the barrel exerts a constant friction force of 0.032 2 N on the ball.
Question:
1. With what speed does the projectile leave the barrel of the cannon?
m/s
2. At what point does the ball have maximum speed?